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Weak and strong convergence theorems of common fixed points for a pair of nonexpansive and asymptotically nonexpansive mappings. (English) Zbl 1099.47057

Let \(K\) be a subset of a Banach space. A mapping \(T:K\to K\) is said to be asymptotically nonexpansive if there exists a sequence \(\{k_n\}\) in \([1,\infty)\) such that \(\lim_n k_n=1\) and \(\| T^nx-T^ny\| \leq k_n\| x-y\| \) for all \(n\in\mathbb{N}\) and for all \(x,y\in K\). In particular, if \(k_n\equiv1\), we call \(T\) a nonexpansive mapping. In this paper, the authors introduce a new three-step iteration method with error terms for a pair of nonexpansive and asymptotically nonexpansive mappings. They obtain some strong and weak convergence theorems for the iteration whenever the space is uniformly convex. An example is included to show that their result substantially extends the corresponding results of S. S.Chang [Indian J.Pure Appl. Math. 32, No. 9, 1297–1307 (2001; Zbl 1034.47039)], Z.–Q.Liu and S. M.Kang [Acta Math.Sin., Engl.Ser.20, No. 6, 1009–1018 (2004; Zbl 1098.47059)] and M. O.Osilike and S. C.Aniagbosor [Math.Comput.Modelling 32, No. 10, 1181–1191 (2000; Zbl 0971.47038)].

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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References:

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